Gaussian Elimination: Solving Systems of Linear Equations

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Discussion Overview

The discussion centers around the use of Gaussian elimination in computer software for solving systems of linear equations, exploring its applications, alternatives, and related matrix decomposition methods.

Discussion Character

  • Technical explanation, Debate/contested

Main Points Raised

  • One participant asserts that Gaussian elimination is indeed used in computer software to solve systems of linear equations.
  • Another participant confirms this but adds that while Gaussian elimination is common, it is not the only method available, suggesting that other techniques may be more suitable in specific scenarios.
  • Some participants mention that methods like conjugate gradients are advantageous for sparse matrices, and matrix decomposition techniques can be beneficial when solving for multiple right-hand sides.
  • LU decomposition is highlighted as being based on Gaussian elimination, while other methods such as Cholesky, LDLT, QR, and SVD are noted as alternatives that may not rely on Gaussian elimination.

Areas of Agreement / Disagreement

Participants generally agree that Gaussian elimination is a method used in software for solving linear equations, but multiple competing views exist regarding its effectiveness compared to other methods, and the discussion remains unresolved on which method is superior in various contexts.

Contextual Notes

Participants mention specific conditions under which alternative methods may be preferred, such as the sparsity of the matrix and the need for numerical stability, but do not resolve the implications of these conditions.

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Is it correct that the Gaussian elimination procedure is used in computer software to solve systems of linear equations?
 
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http://www.mathworks.com/access/helpdesk/help/techdoc/math/f4-983672.html
 
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In short: yes (with pivoting for numerical stability), but it's far from the only way, and in certain cases other methods may be better (for example, some method based on conjugate gradients is good when the matrix is sparse, and matrix decomposition is useful for when you want to solve Ax = b for several different b).
 
There are a variety of matrix decompositions that are used in solving sets of linear equations. LU decomposition is based on Gauss elimination:

http://en.wikipedia.org/wiki/LU_decomposition

but other methods (e.g. Cholesky, LDLT, QR, SVD, etc.) may not.
 

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